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NUMBER THEORY and COMBINATORICS SEMINAR UNIVERSITY of LETHBRIDGE Contents PAST SPEAKERS IN THE NUMBER THEORY AND COMBINATORICS SEMINAR OF THE DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE AT THE UNIVERSITY OF LETHBRIDGE http://www.cs.uleth.ca/~nathanng/ntcoseminar/ Organizers: Nathan Ng and Dave Witte Morris (starting Fall 2011) Amir Akbary (Fall 2007 – Fall 2010) Contents ........................... Fall 2020 2 Spring 2014 ....................... 48 ........................ Spring 2020 5 Fall 2013 .......................... 52 ........................... Fall 2019 9 Spring 2013 ....................... 55 ....................... Spring 2019 12 Fall 2012 .......................... 59 .......................... Fall 2018 16 Spring 2012 ....................... 63 ....................... Spring 2018 20 Fall 2011 .......................... 67 .......................... Fall 2017 24 Fall 2010 .......................... 70 ....................... Spring 2017 28 Spring 2010 ....................... 71 .......................... Fall 2016 31 Fall 2009 .......................... 73 ....................... Spring 2016 34 Spring 2009 ....................... 75 .......................... Fall 2015 39 Fall 2008 .......................... 77 ....................... Spring 2015 42 Spring 2008 ....................... 80 .......................... Fall 2014 45 Fall 2007 .......................... 83 Fall 2020 Open problem session Sep 28, 2020 Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Allysa Lumley (Centre de Recherches Math3́matiques, Montréal) Oct 5, 2020 Primes in short intervals: Heuristics and calculations We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length y around x, where y ≪ (log x)2. In particular, we conjecture that the maximum grows surprisingly slowly as y ranges from log x to (log x)2. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville. Elchin Hasanalizade (University of Lethbridge) Oct 19, 2020 Distribution of the divisor function at consecutive integers The purpose of this talk is to discuss the famous Erdös-Mirsky conjecture and problems on consecutive values of multiplicative functions that arise in connection with this conjecture. We’ll show how the results of Goldston et al. can be used to sharpen Hildebrand’s earlier result on a conjecture of Erdös on limit points of the sequence d(n)=d(n + 1). If time permits. we’ll also discuss some open problems. Kubra Benli (University of Georgia) Oct 26, 2020 Prime power residues modulo p Let p be a prime number. For each positive integer k ≥ 2, it is widely believed that the smallest prime that is a kth power residue modulo p should be O(pϵ), for any ϵ > 0. Elliott k−1 +ϵ proved that such a prime is at most p 4 , for each ϵ > 0. In this talk we will discuss the distribution of prime kth power residues modulo p in the range [1; p], with a more emphasis k−1 +ϵ on the subrange [1; p 4 ] for ϵ > 0. 2 NUMBER THEORY AND COMBINATORICS SEMINAR — UNIVERSITY OF LETHBRIDGE 3 Iren Darijani (University of Lethbridge) Nov 2, 2020 Colourings of G-designs A G-decomposition of a graph H consists of a set V of vertices of H together with a set B of subgraphs (called blocks) of H, each isomorphic to G, that partition the edge set of H.A G-design of order n is a G-decomposition of the complete graph Kn on n vertices. A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge. A G-design is said to be k-colourable if its vertex set can be partitioned into k sets (called colour classes) such that no block is monochromatic. It is k-chromatic if it is k-colourable but is not (k − 1)-colourable. The block intersection graph of a G-design with block set B is the graph with B as its vertex set such that two vertices are adjacent if and only if their associated blocks are not disjoint. The chromatic index of a graph H is the least number of colours that enable each edge of H to be assigned a single colour such that adjacent edges never have the same colour. In this talk we will first see some results on the chromatic index of block intersection graphs of Steiner triple systems. A Steiner triple system of order v is a K3-design of order n with n = v. We will then discuss some other results on k-colourings of e-star systems for all k ≥ 2 and e ≥ 3. An e-star is a complete bipartite graph K1;e. An e-star system of order n > 1, is a G-design of order n when G is an e-star. Finally, we will see some other results on k-colourings of path systems for any k ≥ 2. A path system is a G-design when G is a path. Yash Totani (University of Lethbridge) Nov 16, 2020 On the number of representations of an integer as a sum of k triangular numbers The mth r-gonal number is defined by 2 pr(m) = [(r − 2)m − (r − 4)m]=2: The problem of finding the number of representations of an integer as asumof k r-gonal num- bers is well studied, however a lot of interesting related questions are still unresolved. When r = 4, this problem converts to the classical problem of finding the number of representations of an integer as a sum of k squares. In this talk, we discuss the case r = 3 where we study the number of representations of an integer by k triangular (3-gonal) numbers. Let tk(n) be the number of representations of a positive integer n as a sum of k triangular numbers. One of the oldest results in connection with triangular numbers is due to Gauss who proved that t3(n) ≥ 1 for any positive integer n. We discuss some new results in the theory by Atanasov et al. where they provide exact formulas for t4k(n), which are proved mysteriously. Here, we discuss a method, inspired by the work of Dr. Zafer Selcuk Aygin, which can be used to give a more concrete proof for the result mentioned above using the modular forms of even integral weight. We also obtain an extension of the above and provide formulas for t4k+2(n) using the theory of odd integral weight modular forms for Γ0(8). 4 NUMBER THEORY AND COMBINATORICS SEMINAR — UNIVERSITY OF LETHBRIDGE Dave Morris (University of Lethbridge) Nov 23, 2020 Automorphisms of direct products of some circulant graphs The direct product of two graphs X and Y is denoted X ×Y . This is a natural construction, so any isomorphism from X to X0 can be combined with any isomorphism from Y to Y 0 to obtain an isomorphism from X × Y to X0 × Y 0. Therefore, the automorphism group Aut(X × Y ) contains a copy of (Aut X) × (Aut Y ). It is not known when this inclusion is an equality, even for the special case where Y = K2 is a connected graph with only 2 vertices. Recent work of B. Fernandez and A. Hujdurović solves this problem when X is a “circulant” graph with an odd number of vertices (and Y = K2). We will present a short, elementary proof of this theorem. Milad Fakhari (University of Lethbridge) Nov 30, 2020 The Correction Factors in Artin’s Type Problems In 1927, Emil Artin formulated a conjecture for the density δ(g) of primes for which a given h i F× integer g is a primitive root, i.e., g mod p = p . Such prime p does not split completely q in the splitting field of x − g, indicated by Kq, for all primes q j p − 1. For g =6 −1 and non-square, Artin conjectured Y 1 δ(g) = 1 − : [K : Q] p p In 1957, computations, carried by Derrick H. Lehmer and Emma Lehmer, revealed that for some g (for example g = 5) the conjectured density is quite different from the one computed. Artin provided a correction factor to handle this mismatch. In fact,Q the correction factor appears because of the “entanglement” between number fields Kn = pjn Kp for n square-free when the square-free part of g is congruent to 1 modulo 4. In 1967, Hooley proved the modified Artin’s conjecture under the assumption of the generalized Riemann hypothesis. In this talk, we will present the character sums method formulated by Lenstra, Moree and Stevenhagen which gives us an effective way to find the correction factor and the Euler product of the density in Artin’s conjecture and the cyclicity problem for Serre curves. We will observe that this method does not work in some Artin-like problems such as the Kummerian Titchmarsh divisor problem. At the end, we will indicate that how a modification of the character sums method may lead to the correction factor in such problems. Spring 2020 Open problem session Jan 13, 2020 Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Joy Morris (University of Lethbridge) Jan 20, 2020 Regular Representations of Groups A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the groups of symmetries of polygons, logos, or designs. For example, the dihedral group D8 of order 8 is the group of symmetries of a square. However, there are some challenges with this particular example of visualisation, as many people struggle to understand how reflections and rotations interact as symmetries of a square.
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